For me, it was definitely the commercials for the game that made me first notice. The idea of having an anime-esque character fighting alongside Donald and Goofy was intriguing, especially since it involved going to all of those iconic Disney worlds. The fact that it also featured several Final Fantasy characters that I sort of knew about but never had the opportunity (at the time) to play as was just icing on the cake.
I wanted the game as soon as it came out, but I was unable to get it until January 2003. I was 11 going-on-12 at the time. My brother and I were immediately hooked by the opening and innovative gameplay that sort of reminded us of Ocarina of Time and Majora's mask. It took us both a while to beat it (thanks, Hollow Bastion), but when we did, it was one of the biggest accomplishments that we felt we achieved at the time. The game singlehandedly helped us become Final Fantasy fans as well.
Chain of Memories on the Game Boy Advance presented one hell of a learning curve, but we eventually got through both scenarios and were pumped up for the next game. Around that time, we went back to I and got all of the dalmatians and were amazed at the secret ending.
Then came II. We geeked out so hard upon seeing Axel and hearing "The Thirteenth Struggle" again. Auron and Tron being party members was an added bonus. Despite this, what stood out the most was Roxas who would go on to become my favorite character in the entire franchise.
Re: Chain of Memories came out amidst everyone's doubts. Playing the game was fun and the added bonuses were a welcome addition.
Unfortunately, this is where I ended up on a hiatus in the series. My brother fell out of it thanks to the convoluted narrative (and admittedly, I can't entirely blame him) and we just didn't have the cash at hand at the time to get the DS games (although we still had DS's).
Birth by Sleep came around. My PSP was broken at the time and it wouldn't be until 2012 when I got a working model again. My brother was interested and started playing it again and enjoyed the fact that the story was simple enough and had gameplay that forced him to be strategic. When I finally got my hands on the game (this time, Final Mix), I was blown away by how awesome the game was and wanted to plow through it. The game brought so much to the table and is undoubtedly my favorite in the series.
3D came out, but at this point, I still don't have a 3DS. Nevertheless, it had a gripping story and introduced my new favorite villain in the series: Young Master Xehanort.
I.5 was announced not long after I watched the Super Gaming Brothers complete their Let's Play of Kingdom Hearts I. The instant I heard about the game and especially after hearing about the artbook, it was an instant pre-order for me (still kind of disappointed about the artbook, though.
II.5 is now on the rise and I am eager to try my hand at the Lingering Will, the Roxas boss battle, and the Absent Silhouettes.
As the years have gone by, the series has grown on me to the point that it is up there in my list of favorite video game series alongside Mario, Zelda, Final Fantasy, Sonic, and more. These games have helped me in a lot of ways from helping me during my years of depression to connecting with others who love the series. The games hold a very special place in my heart and I can only hope that Kingdom Hearts III will be even better than what we have played thus far.